The model.—Here we present a brief overview of the model, which is described in detail in the SM. As mentioned above, the dynamics of the system is governed by the matrices \(M\)  and \(V(t)\) , and each environment \(\mathcal {E}_{\alpha }\)  is a collection of harmonic oscillators whose masses and frequencies are given by the diagonal matrices \(m_{\alpha }\)  and \(\omega _{\alpha }\) . In turn, the interaction Hamiltonian is \(H_{\mathcal {S}, \mathcal {E}} = \sum _{\alpha }X^{T} C_{\alpha } q_{\alpha }\) , where \(C_{\alpha }\)  contains the coupling constants between \(\mathcal {S}\)  and \(\mathcal {E}_{\alpha }\) . In Ref. [1]} it was shown that the solution of the Heisenberg equations for \(X\)  and \(q_{\alpha }\)  can be expressed as linear combinations of the corresponding Schrödinger operators, where the coefficients are determined by the dressed Green function \(G\)  of \(\mathcal {S}\) , \(G (t ,t^{\prime })\) , which is the retarded solution of
\(M \partial _{t}^{2} G + V_{R} (t) G + \int _{0}^{t} d\tau \gamma (t - \tau ) \partial _{\tau } G (\tau , t^{\prime }) = \delta (t-t^{\prime }).\) 
